clmnegsubdi2 - Metamath Proof Explorer

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Theorem clmnegsubdi2 25336

Description: Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 29-Sep-2021.)

**Hypotheses**
Ref	Expression
clmpm1dir.v	⊢ 𝑉 = (Base‘𝑊)
clmpm1dir.s	⊢ · = ( ·_𝑠 ‘𝑊)
clmpm1dir.a	⊢ + = (+_g‘𝑊)

**Assertion**
Ref	Expression
clmnegsubdi2	⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = (𝐵 + (-1 · 𝐴)))

**Proof of Theorem clmnegsubdi2**
Step	Hyp	Ref	Expression
1		simp1 1154	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ ℂMod)
2		eqid 2760	. . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
3		eqid 2760	. . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
4	2, 3	clmneg1 25313	. . . 4 ⊢ (𝑊 ∈ ℂMod → -1 ∈ (Base‘(Scalar‘𝑊)))
5	4	3ad2ant1 1151	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → -1 ∈ (Base‘(Scalar‘𝑊)))
6		simp2 1155	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉)
7		simpl 488	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ ℂMod)
8	4	adantr 486	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → -1 ∈ (Base‘(Scalar‘𝑊)))
9		simpr 490	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
10		clmpm1dir.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
11		clmpm1dir.s	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝑊)
12	10, 2, 11, 3	clmvscl 25319	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ -1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐵) ∈ 𝑉)
13	7, 8, 9, 12	syl3anc 1398	. . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐵) ∈ 𝑉)
14	13	3adant2 1149	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐵) ∈ 𝑉)
15		clmpm1dir.a	. . . 4 ⊢ + = (+_g‘𝑊)
16	10, 2, 11, 3, 15	clmvsdi 25323	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ (-1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐴 ∈ 𝑉 ∧ (-1 · 𝐵) ∈ 𝑉)) → (-1 · (𝐴 + (-1 · 𝐵))) = ((-1 · 𝐴) + (-1 · (-1 · 𝐵))))
17	1, 5, 6, 14, 16	syl13anc 1399	. 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = ((-1 · 𝐴) + (-1 · (-1 · 𝐵))))
18	10, 11, 15	clmnegneg 25335	. . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉) → (-1 · (-1 · 𝐵)) = 𝐵)
19	18	3adant2 1149	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (-1 · 𝐵)) = 𝐵)
20	19	oveq2d 7430	. 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((-1 · 𝐴) + (-1 · (-1 · 𝐵))) = ((-1 · 𝐴) + 𝐵))
21		clmabl 25300	. . . 4 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Abel)
22	21	3ad2ant1 1151	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ Abel)
23		simpl 488	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → 𝑊 ∈ ℂMod)
24	4	adantr 486	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → -1 ∈ (Base‘(Scalar‘𝑊)))
25		simpr 490	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉)
26	10, 2, 11, 3	clmvscl 25319	. . . . 5 ⊢ ((𝑊 ∈ ℂMod ∧ -1 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝐴 ∈ 𝑉) → (-1 · 𝐴) ∈ 𝑉)
27	23, 24, 25, 26	syl3anc 1398	. . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (-1 · 𝐴) ∈ 𝑉)
28	27	3adant3 1150	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · 𝐴) ∈ 𝑉)
29		simp3 1156	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
30	10, 15	ablcom 19929	. . 3 ⊢ ((𝑊 ∈ Abel ∧ (-1 · 𝐴) ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((-1 · 𝐴) + 𝐵) = (𝐵 + (-1 · 𝐴)))
31	22, 28, 29, 30	syl3anc 1398	. 2 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((-1 · 𝐴) + 𝐵) = (𝐵 + (-1 · 𝐴)))
32	17, 20, 31	3eqtrd 2799	1 ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = (𝐵 + (-1 · 𝐴)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ‘cfv 6533 (class class class)co 7414 1c1 11128 -cneg 11469 Basecbs 17304 +_gcplusg 17345 Scalarcsca 17348 ·_𝑠 cvsca 17349 Abelcabl 19911 ℂModcclm 25293

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-addf 11206 ax-mulf 11207

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-fz 13565 df-seq 14069 df-struct 17242 df-sets 17259 df-slot 17277 df-ndx 17289 df-base 17305 df-ress 17326 df-plusg 17358 df-mulr 17359 df-starv 17360 df-tset 17364 df-ple 17365 df-ds 17367 df-unif 17368 df-0g 17529 df-mgm 18733 df-sgrp 18824 df-mnd 18840 df-grp 19063 df-minusg 19064 df-mulg 19194 df-subg 19249 df-cmn 19912 df-abl 19913 df-mgp 20277 df-ur 20324 df-ring 20377 df-cring 20378 df-subrg 20735 df-lmod 21049 df-cnfld 21589 df-clm 25294

This theorem is used by: (None)

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